Minimal ring topologies
Let \((R, \tau)\) be a topological ring and denote by \(N(\tau)\) the \(\tau\)-closure of \(\{0\}\) in \(R\). The author calls \(\tau\) a minimal ring topology if there is no ring topology \(\sigma\) on \(R\) strictly weaker than \(\tau\) with \(N(\sigma) = N(\tau)\). He presents examples of Hausdorff topological fields which do not admit a weaker Hausdorff minimal ring topology. This solves an open problem in \textit{W. Wiesław} [Topological fields (1988; Zbl 0661.12011), Problem 2, p. 251]. In the case where \(R\) is a Boolean ring and \(\tau\) is a Hausdorff topology several conditions equivalent to the minimality of \(\tau\) are given. They include the following one: \(\tau\) is an order continuous \(s\)-bounded Fréchet-Nikodym (=monotone) topology. The author is also concerned with minimal ring topologies [locally solid group topologies] on rings of simple functions with values in a field [with values in \(\mathbb{R}]\). As an application, he shows that the topology of convergence in measure on the space of real-valued measurable functions is a minimal ring topology and a minimal locally solid group topology (Corollary 5.7). The first part of the last assertion sharpens a result of \textit{N. T. Peck} [Math. Ann. 178, 209-218 (1968; Zbl 0159.00416), Theorem 1.2]. \{Reviewer's remarks: (1) The proof of Theorem 4.2, \((4) \Rightarrow (7)\), is incomplete since \(F_\gamma\) may be empty. The author has explained to the reviewer how to fill this gap. Moreover, in the same proof, an appeal to the ``if'' part of Theorem 2.8 in \([W3]\) has been inadvertently omitted. The latter result also appears, in a more general form, in \textit{T. A. Sarymsakov} et al. [Ordered algebras (Russian) (1983; Zbl 0542.46001), Proposition I.7.1]. (2) For results closely related to the material of Section 4 and Corollary 5.7 see \textit{C. D. Aliprantis} and \textit{O. Burkinshaw} [Locally solid Riesz spaces (1978; Zbl 0402.46005), Section 24; Ill. J. Math. 24, 164-172 (1980; Zbl 0407.46006)], and \textit{I. Labuda} [Math. Z. 195, 179-196 (1987; Zbl 0601.46006)]. (3) The preprint by \textit{P. A. Abramovich} and the reviewer referred to in the paper under review has appeared in the meantime [Comment. Math. Univ. Carolinae 31, 627-635 (1990; Zbl 0759.46003)].
scientific article; zbMATH DE number 710358